Weak limit of iterates of some random-valued functions and its application

Given a probability space (Ω,A, P), a complete and separable metric space X with the σ-algebra B of all its Borel subsets, a B ⊗A-measurable and contractive in mean f : X × Ω → X, and a Lipschitz F mapping X into a separable Banach space Y we characterize the solvability of the equation ϕ(x) = Ω ϕ (f(x, ω)) P(dω) + F(x) in the class of Lipschitz functions ϕ : X → Y with the aid of the weak limit πf of the sequence of iterates (fn(x, ·))n∈N of f, defined on X × ΩN by f0(x, ω) = x and fn(x, ω) = f fn−1(x, ω), ωn for n ∈ N, and propose a characterization of πf for some special rvfunctions in Hilbert spaces